That's actually a good question. I don't know if you gain much from that objection besides clarity. After all, both "kinds" of probabilities share the same math

That's actually a good question. I don't know if you gain much from that objection besides clarity. After all, both "kinds" of probabilities share the same mathematics. I think it's the other way around. People have certain intuitions, or prejudices if you wish, and it's useful to have terms that match your intuitions - even if they are slightly redundant. I find it helps me understand and explain statistical problems better. About the coin: I think you have to distinguish between one concrete flip of the coin, and the coin itself. The coin itself has certain probabilities for generating heads and tails, and they don't change. A concrete flipping of the coin is a different situation. You might see the coin, and know with near 100% certainty what it shows. Or you might have some machine which detects whether it is heads or tails, and this machine has a certain accuracy. When it signals heads, you might say it has a 95% plausibility to really be heads. And before you measured the state of the coin, you might say the plausibility is 50%, assuming the coin is fair. So the plausibility does change at time of observation (or when you get new information). What about the time of the flip? Well, after the flip has occurred, this concrete instance is either 100% heads or 100% tails of course. If I repeat this exact coin toss (which I can't do in reality), I will always get the same result. And the probability to have heads, when you have heads, is 100% (P(toss #1 is heads | toss #1 is heads) = 100%). This statement is a bit silly, but I think it shows that the probability of a thrown, lying coin to be heads or tails is not a very interesting quantity. Our degree of belief, given certain facts, that it is heads or tails, is interesting. And the probability of the coin in general, determined by its geometry, is interesting. And especially interesting is how you get from one to the other, which is where Bayesian inference, hypothesis testing, and so on come in (The coin is known to be rigged with heads=99%. My flawed detector says it is tails. What should I believe is the state of the coin, or what should I bet on?). Finally, I should note that you should take what I've said with a grain of salt. I'm at best an armchair statistician, although I do get to think about stuff like this a lot at work (as a physicist).